# Breaking the formulas
---
## How to write a complex formula in TAAL?
$$\frac{(123 + 420) \cdot (560 - 32) - 276 \cdot 4}{765 + 187}$$
---
## Step \#1
Our goal is to translate the infix form to the prefix.
Find the latest action in the math expression!
----
$$\frac{\color{#900}{(123 + 420) \cdot (560 - 32) - 276 \cdot 4}}{\color{#00c}{765 + 187}}$$
In our case, it is a **division**.
$$\frac{\color{#900}{<top\ section>}}{\color{#00c}{<bottom\ section>}}$$
Never mind about numerator and denominator now!
---
## Step \#2
Write the found action in TAAL.
```lisp
(/ <top section> <bottom section>)
```
We just have excluded one action from our complex formula. But we need to go forward.
---
## The repetition loop of two steps
To get the answer we need to parse the top and bottom sections step by step. Begin from a bottom _(because it is more simple, we have a plain denominator)_.
$$765 + 187$$
It is just a sum of two elements. Write it in TAAL.
```lisp
(/ <top section> (+ 765 187))
```
----
Take a look at the top section now.
$$(123 + 420) \cdot (560 - 32) - 276 \cdot 4$$
Now just repeat Step \#1 and find the latest action for the part of the formula.
----
As you can see it is **subtraction**.
$$<left> - <right>$$
As in the previous step do not figure out about the left and the right.
----
Just repeat Step \#2 and improve our formula in TAAL.
```lisp
(/ (- <left> <right>) (+ 765 187))
```
Now we may simplify the part of the top section.
----
The first look at the \<right\> placeholder says us to simplify it first.
So...
$$276 \cdot 4$$
becomes
```lisp
(* 276 4)
```
and inserts in the formula.
```lisp
(/ (- <left> (* 276 4)) (+ 765 187))
```
----
We strait see the finish line. Just do both steps one more time and have the result.
$$(123 + 420) \cdot (560 - 32)$$
The last action here is **multiplication**. So we have...
```lisp
(* <sum> <subtraction>)
```
----
Put it to TAAL formula again.
```lisp
(/ (- (* <sum> <subtraction>) (* 276 4)) (+ 765 187))
```
The last step is absolutely obvious. Just do it.
----
$$123 + 420$$
```lisp
(+ 123 420)
```
$$560 - 32$$
```lisp
(- 560 32)
```
### The final
```lisp
(/ (- (* (+ 123 420) (- 560 32)) (* 276 4)) (+ 765 187))
```
---
## Complete result
$$\frac{(123 + 420) \cdot (560 - 32) - 276 \cdot 4}{765 + 187}$$
```lisp
(/ (- (* (+ 123 420) (- 560 32)) (* 276 4)) (+ 765 187))
```
If you will put the formula to the interpreter and execute it you see the answer **300**.
Well done!
----
## All steps together.
$$\frac{(123 + 420) \cdot (560 - 32) - 276 \cdot 4}{765 + 187}$$
```lisp
(/ <top section> <bottom section>)
(/ <top section> (+ 765 187))
(/ (- <left> <right>) (+ 765 187))
(/ (- <left> (* 276 4)) (+ 765 187))
(/ (- (* <sum> <subtraction>) (* 276 4)) (+ 765 187))
(/ (- (* (+ 123 420) (- 560 32)) (* 276 4)) (+ 765 187))
```
----
To see and better understand the TAAL prefix notation just look at the formula again in the formatted view.
```lisp=
(/
(-
(*
(+ 123 420)
(- 560 32)
)
(* 276 4)
)
(+ 765 187)
)
```
---
## Control tasks
$$\frac{4+6}{5\cdot2}+3\cdot\frac{5+5}{1+\frac{1}{2}} \tag{1}$$
$$\sqrt{\left(\frac{4\cdot2+1}{1+2}\right)^2+\left(\sqrt{2\cdot(3+5)}\right)^2} \tag{2}$$
----
## Answers
$$\frac{4+6}{5\cdot2}+3\cdot\frac{5+5}{1+\frac{1}{2}} = 21 \tag{1}$$
$$\sqrt{\left(\frac{4\cdot2+1}{1+2}\right)^2+\left(\sqrt{2\cdot(3+5)}\right)^2} = 5 \tag{2}$$
---
# The end.
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